For a Third Person game, how would I go about making the character's arm follow the camera?
It really depends on how you are controlling your character and camera; there are many different methods (ie. camera actuator, edit object actuator -> mode: track to)

The way I would go about setting up controls like Just Cause 2 (shooting) is to add an empty for the location of the camera behind the character, and another empty (or object) for a reciticle and when holding down an "aiming" button move the camera to the location of the empty and "track to" the reciticle with both the camera and the arm/gun. And make the mouse control the reciticle.